- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤ (values match St000056The decomposition (or block) number of a permutation.)

Values

[1,0] => [1] => 0

[1,0,1,0] => [1,2] => 1

[1,1,0,0] => [2,1] => 0

[1,0,1,0,1,0] => [1,2,3] => 2

[1,0,1,1,0,0] => [1,3,2] => 1

[1,1,0,0,1,0] => [2,1,3] => 1

[1,1,0,1,0,0] => [2,3,1] => 0

[1,1,1,0,0,0] => [3,2,1] => 0

[1,0,1,0,1,0,1,0] => [1,2,3,4] => 3

[1,0,1,0,1,1,0,0] => [1,2,4,3] => 2

[1,0,1,1,0,0,1,0] => [1,3,2,4] => 2

[1,0,1,1,0,1,0,0] => [1,3,4,2] => 1

[1,0,1,1,1,0,0,0] => [1,4,3,2] => 1

[1,1,0,0,1,0,1,0] => [2,1,3,4] => 2

[1,1,0,0,1,1,0,0] => [2,1,4,3] => 1

[1,1,0,1,0,0,1,0] => [2,3,1,4] => 1

[1,1,0,1,0,1,0,0] => [2,3,4,1] => 0

[1,1,0,1,1,0,0,0] => [2,4,3,1] => 0

[1,1,1,0,0,0,1,0] => [3,2,1,4] => 1

[1,1,1,0,0,1,0,0] => [3,2,4,1] => 0

[1,1,1,0,1,0,0,0] => [3,4,2,1] => 0

[1,1,1,1,0,0,0,0] => [4,3,2,1] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 4

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 3

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 3

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 3

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 2

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 1

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 1

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 1

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 3

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 1

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 1

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 2

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 1

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 0

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 1

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 0

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 0

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 0

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 1

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 0

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 1

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 1

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 5

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 3

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 3

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 3

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => 2

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 2

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 4

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 3

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 3

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 3

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 1

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 1

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 1

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => 2

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 1

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + 2\ q + q^{2}$

$F_{4} = 5 + 5\ q + 3\ q^{2} + q^{3}$

$F_{5} = 14 + 14\ q + 9\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 42 + 42\ q + 28\ q^{2} + 14\ q^{3} + 5\ q^{4} + q^{5}$

Description

The number of global ascents of a permutation.
The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$
Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$
For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives oeis:A003319.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).